On the Collatz-Wielandt sets associated with a cone-preserving map

Abstract

Let K be a closed, pointed, full cone in R n . In their treatment of Perron-Frobenius theory for a linear map A preserving K by Wielandt's approach, Barker and Schneider introduced four sets, namely, Ω, Ω 1, Σ, and Σ 1, the Collatz-Wielandt sets associated with A. We determine the greatest lower bound and the least upper bound of these sets. Some known results for a nonnegative matrix relating its spectral radius to its upper and lower Collatz-Wielandt number are extended to the setting of a cone-preserving map. Applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered.